A Tannakian Description for Parahoric Bruhat-Tits Group Schemes

Abstract

Let K be a field which is complete with respect to a discrete valuation and let O be the ring of integers in K. We study the Bruhat-Tits building B(G) and the parahoric Bruhat-Tits group schemes G_F associated to a connected reductive split linear algebraic group G defined over O.

In order to study these objects we use the theory of Tannakian duality, developed by Saavedra Rivano, which shows how to recover G from its category of finite rank projective representations over O. We also use Moy-Prasad filtrations in order to define lattice chains in any such representation. Using these two tools, we give a Tannakian description to B(G).

We also define a functor Aut_F associated to a facet F in B(G) in terms of lattice chains in a Tannakian way. We show that Aut_F is representable by an affine group scheme of finite type, has the same generic fiber as G, and satisfies Aut_F(O_E) = G_F (O_E) for every unramified Galois extension E of K.

Moreover, we show that there is a canonical morphism from G_F to Aut_F, which we conjecture to be an isomorphism. We prove that it is an isomorphism when the residue characteristic of K is zero and G is arbitrary, when G = GL_n and K is arbitrary, and when F is the minimal facet containing the origin and G and K are arbitrary.